A343058 -id:A343058 - cp's OEIS frontend

A178817 Decimal expansion of the area of the regular 7-gon (heptagon) of edge length 1.

3, 6, 3, 3, 9, 1, 2, 4, 4, 4, 0, 0, 1, 5, 8, 8, 9, 9, 2, 5, 3, 6, 1, 9, 3, 0, 0, 7, 6, 0, 0, 2, 2, 0, 5, 7, 8, 7, 3, 5, 0, 1, 0, 3, 6, 1, 5, 9, 5, 4, 4, 4, 9, 1, 7, 1, 4, 5, 9, 8, 0, 4, 0, 9, 5, 1, 0, 2, 9, 9, 8, 5, 2, 3, 6, 3, 0, 4, 6, 0, 0, 5, 5, 6, 2, 7, 3, 0, 7, 1, 5, 2, 9, 5, 8, 1, 0, 8, 9, 4, 3, 7, 1, 0, 4

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 16 2010

			3.63391244400158899253619300760022057873501036159544491714598040951029...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Heptagon
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 6

Cf. A178818, A343058.

Cf. Areas of other regular polygons: A120011, A102771, A104956, A090488, A256853, A178816, A256854, A178809.

SetDefaultRealField(RealField(100)); R:=RealField(); 7*Cot(Pi(R)/7)/4; // G. C. Greubel, Jan 22 2019

evalf[120]((7/4)*cot(Pi/7)); # Muniru A Asiru, Jan 22 2019

Mathematica
```
RealDigits[7*Cot[Pi/7]/4, 10, 100][[1]]
```

p=7; a=(p/4)*cotan(Pi/p)  \\ Set realprecision in excess. - Stanislav Sykora, Apr 12 2015

numerical_approx(7*cot(pi/7)/4, digits=100) # G. C. Greubel, Jan 22 2019

Equals (7/4) * cot(Pi/7).
From Michal Paulovic, Dec 27 2022: (Start)
Equals 7 / (4 * tan(Pi/7)) = 7 / (4 * A343058).
Equals sqrt(7/3 * (35 + 2 * 196^(1/3) * ((13 - 3 * sqrt(3) * i)^(1/3) + (13 + 3 * sqrt(3) * i)^(1/3)))) / 4.
Equals sqrt(7/4) * sqrt(35/12 + (637/54 - sqrt(-2401/108))^(1/3) + (637/54 + sqrt(-2401/108))^(1/3)).
(End)
A root of the polynomial 4096*x^6 - 62720*x^4 + 115248*x^2 - 16807. - Joerg Arndt, Jan 02 2023

A374971 Decimal expansion of the apothem (inradius) of a regular heptagon with unit side length.

Original entry on oeis.org

1, 0, 3, 8, 2, 6, 0, 6, 9, 8, 2, 8, 6, 1, 6, 8, 2, 8, 3, 5, 8, 1, 7, 6, 9, 4, 3, 0, 7, 4, 2, 9, 2, 0, 1, 6, 5, 3, 5, 2, 8, 6, 0, 1, 0, 3, 3, 1, 2, 9, 8, 4, 2, 6, 2, 0, 4, 1, 7, 0, 8, 6, 8, 8, 4, 3, 1, 5, 1, 4, 2, 4, 3, 5, 3, 2, 2, 9, 8, 8, 5, 8, 7, 3, 2, 2, 0, 8, 7, 7

Offset: 1

Paolo Xausa, Jul 26 2024

			1.0382606982861682835817694307429201653528601033...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Wikipedia, Apothem.
Index entries for algebraic numbers, degree 6.

Cf. A374957 (circumradius), A374972 (sagitta), A178817 (area).
Cf. apothem of other polygons with unit side length: A020769 (triangle), A020761 (square), A375067 (pentagon), A010527 (hexagon), A174968 (octagon), A375152 (9-gon), A179452 (10-gon), A375191 (11-gon), A375193 (12-gon).
Cf. A178818, A073052, A343058.

Mathematica
```
First[RealDigits[Cot[Pi/7]/2, 10, 100]]
```

 5/(Pi/7) \\ Charles R Greathouse IV, Feb 05 2025

Equals cot(Pi/7)/2 = A178818/2.
Equals 1/(2*tan(Pi/7)) = 1/(2*A343058).
Equals A374957*cos(Pi/7) = A374957*A073052.
Equals A374957 - A374972.
Largest of the 6 real-valued roots of 448*x^6 -560*x^4 +84*x^2 -1 =0. - R. J. Mathar, Aug 29 2025

A381153 Decimal expansion of the isoperimetric quotient of a regular heptagon.

Original entry on oeis.org

9, 3, 1, 9, 4, 0, 6, 2, 3, 4, 9, 9, 0, 9, 5, 7, 4, 5, 9, 5, 2, 2, 2, 6, 3, 0, 0, 8, 9, 4, 2, 2, 7, 5, 4, 5, 7, 4, 5, 2, 8, 5, 2, 5, 1, 5, 4, 7, 1, 5, 3, 1, 5, 6, 1, 2, 7, 3, 2, 0, 2, 2, 6, 8, 8, 6, 4, 5, 2, 5, 3, 9, 4, 8, 0, 5, 4, 7, 8, 5, 6, 9, 3, 7, 7, 2, 8, 6, 7, 1

Offset: 0

Paolo Xausa, Feb 15 2025

For the definition of isoperimetric quotient, see A381152.

			0.93194062349909574595222630089422754574528525154715...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Isoperimetric Quotient.
Wikipedia, Isoperimetric inequality.

Cf. A178817, A343058.

Cf. isoperimetric quotient of other regular polygons: A073010 (triangle), A003881 (square), A381152 (pentagon), A093766 (hexagon), A196522 (octagon), A381154 (9-gon), A381155 (10-gon), A381156 (11-gon), A381157 (12-gon).

First[RealDigits[Pi/(7*Tan[Pi/7]), 10, 100]]

Equals Pi/(7*tan(Pi/7)) = Pi/(7*A343058).

Equals (4/49)*Pi*A178817.

A343057 Decimal expansion of tan(Pi/32).

Original entry on oeis.org

0, 9, 8, 4, 9, 1, 4, 0, 3, 3, 5, 7, 1, 6, 4, 2, 5, 3, 0, 7, 7, 1, 9, 7, 5, 2, 1, 2, 9, 1, 3, 2, 7, 4, 3, 2, 2, 9, 3, 0, 5, 2, 4, 5, 0, 6, 9, 9, 2, 0, 2, 6, 9, 5, 9, 8, 0, 9, 1, 6, 1, 2, 1, 1, 3, 4, 4, 1, 9, 4, 3, 8, 7, 3, 0, 8, 1, 2, 9, 7, 2, 2, 5, 6, 4, 8, 5, 2, 1, 4, 1, 8, 0, 3, 7, 3, 6, 0, 0, 1, 3, 7, 0, 6, 7, 1, 6, 9, 7, 7, 9, 1, 7, 6, 5

Offset: 0

Seiichi Manyama, Apr 04 2021

			0.098491403357164253077197...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 8.

Cf. A343055 (sin(Pi/32)), A343056 (cos(Pi/32)).

tan(Pi/m): A002194 (m=3), A019934 (m=5), A020760 (m=6), A343058 (m=7), A188582 (m=8), A019918 (m=9), A019916 (m=10), A019913 (m=12), A343059 (m=14), A019910 (m=15), A343060 (m=16), A343061 (m=17), A019908 (m=18), A019907 (m=20), A343062 (m=24), A019904 (m=30), A343057 (m=32), A019903 (m=36).

R:= RealField(125); Tan(Pi(R)/32); // G. C. Greubel, Sep 30 2022

RealDigits[Tan[Pi/32], 10, 120, -1][[1]] (* Amiram Eldar, Apr 27 2021 *)

PARI
```
tan(Pi/32)
```

sqrt((2-sqrt(2+sqrt(2+sqrt(2))))/(2+sqrt(2+sqrt(2+sqrt(2)))))

numerical_approx(tan(pi/32), digits=125) # G. C. Greubel, Sep 30 2022

Equals sqrt( (2-sqrt(2+sqrt(2+sqrt(2))))/(2+sqrt(2+sqrt(2+sqrt(2)))) ).

cp's OEIS Frontend

A178817 Decimal expansion of the area of the regular 7-gon (heptagon) of edge length 1.

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A374971 Decimal expansion of the apothem (inradius) of a regular heptagon with unit side length.

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A381153 Decimal expansion of the isoperimetric quotient of a regular heptagon.

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A343057 Decimal expansion of tan(Pi/32).

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